Stochastic hydrodynamical limits of particle systems

Abstract

Even small noise can have substantial influence on the dynamics of differential equations, e.g. for nucleation/coarsening and interface dynamics in phase transformations. The aim of this work is to establish accurate models for the noise in macroscopic differential equations, related to phase transformations/reactions, derived from more fundamental microscopic master equations. For this purpose the mathematical paradigm of the dynamic Ising model is considered in the relatively tractable case of stochastic spin flip dynamics and long range spin/spin interactions. More specifically, this paper shows that localized spatial averages, with width epsilon, of solutions to such Ising systems with long range interaction of range O(1), are approximated with error O(epsilon + (gamma/epsilon)(2d)) in distribution by a solution of an Ito stochastic differential equation, with drift as in the corresponding mean field model and a small diffusion coefficient of order (gamma/epsilon)(d/2), generating noise with spatial correlation length epsilon, where gamma is the distance between neighboring spin sites on a uniform periodic lattice in R-d. To determine the correct noise is subtle in the sense that there are expected values, i.e. observables, that require different noise: the expected values that can be accurately approximated by the Einstein-diffusion and the expected values that need an alternative diffusion related to large deviation theory are identified; for instance dendrite dynamics up to a bounded time needs Einstein diffusion while transition rates need a different diffusion model related to invariant measures. The elementary proofs use O((gamma/epsilon)(2d)) consistency of the Kolmogorov-backward equations for the averaged spin and the stochastic differential equation and show that the long range interaction yields smoothing, which contributes with the O(epsilon) error. A new aspect of the derivation is that the error, based on residuals and weights, is computable and suitable for adaptive refinements and modeling.